33 research outputs found
Group Analysis of Nonlinear Fin Equations
Group classification of a class of nonlinear fin equations is carried out
exhaustively. Additional equivalence transformations and conditional
equivalence groups are also found. They allow to simplify results of
classification and further applications of them. The derived Lie symmetries are
used to construct exact solutions of truly nonlinear equations for the class
under consideration. Nonclassical symmetries of the fin equations are
discussed. Adduced results amend and essentially generalize recent works on the
subject [M. Pakdemirli and A.Z. Sahin, Appl. Math. Lett., 2006, V.19, 378-384;
A.H. Bokhari, A.H. Kara and F.D. Zaman, Appl. Math. Lett., 2006, V.19,
1356-1340].Comment: 6 page
Potential Nonclassical Symmetries and Solutions of Fast Diffusion Equation
The fast diffusion equation is investigated from the
symmetry point of view in development of the paper by Gandarias [Phys. Lett. A
286 (2001) 153-160]. After studying equivalence of nonclassical symmetries with
respect to a transformation group, we completely classify the nonclassical
symmetries of the corresponding potential equation. As a result, new wide
classes of potential nonclassical symmetries of the fast diffusion equation are
obtained. The set of known exact non-Lie solutions are supplemented with the
similar ones. It is shown that all known non-Lie solutions of the fast
diffusion equation are exhausted by ones which can be constructed in a regular
way with the above potential nonclassical symmetries. Connection between
classes of nonclassical and potential nonclassical symmetries of the fast
diffusion equation is found.Comment: 13 pages, section 3 is essentially revise
Group Analysis of Variable Coefficient Diffusion-Convection Equations. I. Enhanced Group Classification
We discuss the classical statement of group classification problem and some
its extensions in the general case. After that, we carry out the complete
extended group classification for a class of (1+1)-dimensional nonlinear
diffusion--convection equations with coefficients depending on the space
variable. At first, we construct the usual equivalence group and the extended
one including transformations which are nonlocal with respect to arbitrary
elements. The extended equivalence group has interesting structure since it
contains a non-trivial subgroup of non-local gauge equivalence transformations.
The complete group classification of the class under consideration is carried
out with respect to the extended equivalence group and with respect to the set
of all point transformations. Usage of extended equivalence and correct choice
of gauges of arbitrary elements play the major role for simple and clear
formulation of the final results. The set of admissible transformations of this
class is preliminary investigated.Comment: 25 page